The pdf of \(Z_i=X_i-\theta _i\)  is \(f_i(z),\; i=1,2.\)  To introduce dependence between random variables \(X_1\)  and \(X_2\) , consider Farlie-Gumbel-Morgenstern family copula (\(C(u,v)=uv-(1-u)(1-v)uv,\;0\le u\le v\le 1\) ; see, page 77 of [1]}) providing the joint pdf of \(\underline{Z}=(Z_1,Z_2)\)  as
\(f(z_1,z_2)=2e^{-z_1}e^{-z_2}\left[e^{-z_1} +e^{-z_2}-2e^{-z_1}e^{-z_2}\right],\;z_1>0,\;z_2>0 .\) 
